I'm not a native speaker of Japanese, but I hope my answer could help! I figure you already know that くれる always needs てform of verbs (except for when くれる is used as the verb "give". いて is a て form of いる (be / exist), and an adjective alone cannot be linked with くれる directly, so it requires ~ている to be linked with くれる. So いてくれる can be used in a situation when someone is being in a certain state/condition for the sake of someone else (instead of doing something for someone which is the common way of using くれる). I don't really find it weird but it might sound weird from English speakers' point of view. ex）好きでいてくれてありがとう Thank you for being 好き. (Thank you for loving me)

Here you are! Hope this helps you. 1 私が落ち込んでいると、猫のキティはいつもそばにいてくれる。（My cat Kitty always stays near me whenever I'm feeling depressed.) 2 私のお母さんはいつも怒らないでいてくれる。（My mother always stays patient = does not scold me.) 3 レジをすませちゃうから、そこにいてくれる？（Will you stay there while I'll finish off with my bill?) 4 今日はお客様がくるから家にいてくれる？（We're having a guest today, so could you stay at home?) "いて（る）" can be translated into "stay" in most cases, I think. Jnnk's example can also be translated as "Thank you for staying in love with me".